Design of Nonbinary LDPC Cycle Codes with Large Girth from Circulants and Finite Fields

<div>Tanner graph of <svg height="11.7248pt" id="M54" style="vertical-align:-3.291161pt" version="1.1" viewbox="-0.0498162 -8.43364 15.181 11.7248" width="15.181pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M746 650H445V615C520 608 528 602 528 517V370H255V517C255 602 263 608 335 615V650H35V615C112 608 121 602 121 517V133C121 48 112 42 39 35V0H337V35C263 42 255 48 255 133V319H528V133C528 48 520 42 441 35V0H746V35C670 42 662 48 662 132V517C662 602 668 608 746 615V650Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,10.217,3.132)"><path d="M460 334C460 396 434 451 378 451C330 451 241 408 148 300H146L237 679C241 697 241 710 232 710C213 710 153 684 67 675L66 646H95C141 646 145 642 134 595L39 170C23 97 31 54 46 33C64 8 100 -12 137 -12C178 -12 234 3 298 43C391 101 460 222 460 334ZM371 320C371 204 316 89 248 51C230 41 208 37 192 37C143 37 102 72 119 166C124 194 129 215 135 235C202 323 298 392 335 392C353 392 371 372 371 320Z"></path></g></svg> (or <span class="nowrap"><svg height="11.7248pt" id="M55" style="vertical-align:-3.291161pt" version="1.1" viewbox="-0.0498162 -8.43364 19.8577 11.7248" width="19.8577pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M746 650H445V615C520 608 528 602 528 517V370H255V517C255 602 263 608 335 615V650H35V615C112 608 121 602 121 517V133C121 48 112 42 39 35V0H337V35C263 42 255 48 255 133V319H528V133C528 48 520 42 441 35V0H746V35C670 42 662 48 662 132V517C662 602 668 608 746 615V650Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,10.217,3.132)"><path d="M504 89L488 119C453 85 428 69 416 69C408 69 405 75 412 102L462 304C492 438 460 451 436 451S388 441 356 423C311 397 229 336 170 256H168L189 340C208 418 200 451 177 451C144 451 82 413 24 352L40 322C65 345 98 369 108 369C114 369 119 363 112 335L28 -3L36 -12C54 -3 83 4 112 10C125 73 138 122 153 174C205 258 328 382 376 382C395 382 399 369 384 305L330 93C312 25 323 -12 351 -12C378 -12 439 21 504 89Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,14.876,3.132)"><path d="M460 334C460 396 434 451 378 451C330 451 241 408 148 300H146L237 679C241 697 241 710 232 710C213 710 153 684 67 675L66 646H95C141 646 145 642 134 595L39 170C23 97 31 54 46 33C64 8 100 -12 137 -12C178 -12 234 3 298 43C391 101 460 222 460 334ZM371 320C371 204 316 89 248 51C230 41 208 37 192 37C143 37 102 72 119 166C124 194 129 215 135 235C202 323 298 392 335 392C353 392 371 372 371 320Z"></path></g></svg>):</span> (a) <svg height="11.7248pt" id="M56" style="vertical-align:-3.291161pt" version="1.1" viewbox="-0.0498162 -8.43364 15.181 11.7248" width="15.181pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M746 650H445V615C520 608 528 602 528 517V370H255V517C255 602 263 608 335 615V650H35V615C112 608 121 602 121 517V133C121 48 112 42 39 35V0H337V35C263 42 255 48 255 133V319H528V133C528 48 520 42 441 35V0H746V35C670 42 662 48 662 132V517C662 602 668 608 746 615V650Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,10.217,3.132)"><path d="M460 334C460 396 434 451 378 451C330 451 241 408 148 300H146L237 679C241 697 241 710 232 710C213 710 153 684 67 675L66 646H95C141 646 145 642 134 595L39 170C23 97 31 54 46 33C64 8 100 -12 137 -12C178 -12 234 3 298 43C391 101 460 222 460 334ZM371 320C371 204 316 89 248 51C230 41 208 37 192 37C143 37 102 72 119 166C124 194 129 215 135 235C202 323 298 392 335 392C353 392 371 372 371 320Z"></path></g></svg> over GF (2); (b) <svg height="11.7248pt" id="M57" style="vertical-align:-3.291161pt" version="1.1" viewbox="-0.0498162 -8.43364 19.8577 11.7248" width="19.8577pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M746 650H445V615C520 608 528 602 528 517V370H255V517C255 602 263 608 335 615V650H35V615C112 608 121 602 121 517V133C121 48 112 42 39 35V0H337V35C263 42 255 48 255 133V319H528V133C528 48 520 42 441 35V0H746V35C670 42 662 48 662 132V517C662 602 668 608 746 615V650Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,10.217,3.132)"><path d="M504 89L488 119C453 85 428 69 416 69C408 69 405 75 412 102L462 304C492 438 460 451 436 451S388 441 356 423C311 397 229 336 170 256H168L189 340C208 418 200 451 177 451C144 451 82 413 24 352L40 322C65 345 98 369 108 369C114 369 119 363 112 335L28 -3L36 -12C54 -3 83 4 112 10C125 73 138 122 153 174C205 258 328 382 376 382C395 382 399 369 384 305L330 93C312 25 323 -12 351 -12C378 -12 439 21 504 89Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,14.876,3.132)"><path d="M460 334C460 396 434 451 378 451C330 451 241 408 148 300H146L237 679C241 697 241 710 232 710C213 710 153 684 67 675L66 646H95C141 646 145 642 134 595L39 170C23 97 31 54 46 33C64 8 100 -12 137 -12C178 -12 234 3 298 43C391 101 460 222 460 334ZM371 320C371 204 316 89 248 51C230 41 208 37 192 37C143 37 102 72 119 166C124 194 129 215 135 235C202 323 298 392 335 392C353 392 371 372 371 320Z"></path></g></svg> over GF (4); (c) Tanner graph.</div>

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Figure 1

Figure 1: Design of Nonbinary LDPC Cycle Codes with Large Girth from Circulants and Finite Fields 